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In a topological abelian group, convergence of a series is defined as convergence of the sequence of partial sums. An important concept when considering series is unconditional convergence, which guarantees that the limit of the series is invariant under permutations of the summands.
The purpose of this article is to serve as an annotated index of various modes of convergence and their logical relationships. For an expository article, see Modes of convergence. Simple logical relationships between different modes of convergence are indicated (e.g., if one implies another), formulaically rather than in prose for quick ...
That the series formed by calculating the expected value of the outcome's distance from a particular value may converge to 0; That the variance of the random variable describing the next event grows smaller and smaller. These other types of patterns that may arise are reflected in the different types of stochastic convergence that have been ...
Unconditional convergence is often defined in an equivalent way: A series is unconditionally convergent if for every sequence () =, with {, +}, the series = converges. If X {\displaystyle X} is a Banach space , every absolutely convergent series is unconditionally convergent, but the converse implication does not hold in general.
A series can be uniformly convergent and absolutely convergent without being uniformly absolutely-convergent. For example, if ƒ n (x) = x n /n on the open interval (−1,0), then the series Σf n (x) converges uniformly by comparison of the partial sums to those of Σ(−1) n /n, and the series Σ|f n (x)| converges absolutely at each point by the geometric series test, but Σ|f n (x)| does ...
In numerical analysis, Aitken's delta-squared process or Aitken extrapolation is a series acceleration method used for accelerating the rate of convergence of a sequence. It is named after Alexander Aitken, who introduced this method in 1926. [1] It is most useful for accelerating the convergence of a sequence that is converging linearly.
The abscissa, line and half-plane of convergence of a Dirichlet series are analogous to radius, boundary and disk of convergence of a power series. On the line of convergence, the question of convergence remains open as in the case of power series. However, if a Dirichlet series converges and diverges at different points on the same vertical ...
In mathematics, a series acceleration method is any one of a collection of sequence transformations for improving the rate of convergence of a series. Techniques for series acceleration are often applied in numerical analysis , where they are used to improve the speed of numerical integration .